A time-varying transmissibility allows to quantify the effects of interventions and changes in the behaviour of people, in real-time. This is of key importance to policy makers, as the interventions often have immense societal and health costs. Understanding, while an epidemic develops, whether the implemented interventions are sufficient or not, or if interventions could be lifted, is essential. The Norwegian government's strategy was to control the epidemic, and this was achieved by multiple national and local non-pharmaceutical interventions, which are reflected in the temporal variations of the reproduction number. Our approach is the first which allows to monitor a daily-varying reproduction number when using a complex compartmental model informed by multiple streams of data.
The fact that our estimates of \(R_t\)  react rapidly to changes in the test data,

means that the situation is captured only with a delay given by the generation time of the disease under study and the time gap between transmission and testing. For Covid-19, this amounts to about a week, because of a generation time of about 5 days and a delay between transmission and test of about 2 days.
Picking up an exponential growth (\(R_t>1\) ) before the epidemic grows out of control is essential for surveillance. The possibility of our method to validate the efficacy of contact tracing, to lead back \(R_t\)  to below 1, or not, is also very important.
We have shown how daily reproduction numbers and the latent compartment-wise populations in an SEIR model can be put into a state space model, so that an SMC technique for inference can be used. Obtaining unbiased estimates of the marginal likelihoods also makes it possible to do parameter estimation within a particle MCMC framework, although more work is needed here to make this computationally efficient. Compared to a parallel effort using Approximate Bayesian Computation (ABC) [1]}, the SMC approach is much faster and also easier to modify with respect to model changes, confirming the findings in [2]}. Our implementation is modular, does not depend on the specific epidemic model (here SEIR), so that alternatives can easily be tested.
So far only simple bootstrap filters have been applied. This can be improved by utilising more efficient proposals, alternative algorithms such as the resample-move algorithm [3]} which was used in [2]} or the recent promising ideas of iterated auxiliary filters and twisted models [5]}, [6]}. We expect these approaches to be very useful when we expand the SEIR model to have different reproductions numbers in each of the eleven Norwegian counties.
We compared three simple dynamics for \(R_t\)  and found that the autoregressive model was slightly better than the others. More work needs to be done here to compare the models in terms of prediction. In our approach it is easy to predict the future hospitalisation incidence and the number of positive cases tested. Note however that such simple dynamic models for \(\lbrace R_t\rbrace \)  are mostly suitable for now-casting and for short term forecasting, because of the lack of stationary that interventions and feedbacks imply.
There are then interesting questions on how to use our probabilistic prediction of the time varying transmission strength to propagate uncertainty, in the context of variable planned interventions.

Estimation of (static) parameters is a challenging task in SMC. Several parameters related to the SEIR model, as well as parameters related to the observation processes, were pre-estimated based on external data sources. In [1]}, we used a version of ABC. Parameters in the \(\lbrace R_t\rbrace \)  dynamic process were estimated online based on the procedure of sufficient statistics, a method that can lead to degeneracy. We have also tested out the particle Metropolis-Hastings algorithm by [8]}. However convergence was slow and challenging, because of the computational cost of running the SMC algorithm even for a small number of particles. We validated our estimates in the supplementary material, and find that our online estimation procedure worked reasonably well for the given models and data. However, some experiments with the AR model when also including estimation of the parameter \(\mu \) , did cause degeneracy problems as have been reported in in [9]}.
A more efficient SMC algorithm might be better in utilising the potential of such algorithms to estimate static parameters. However, our estimates of \(\lbrace R_t\rbrace \)  appear to be relatively robust with respect to changes in these parameters. Other parameters related to the SEIR model and the observation processes can be more important.
Communicating uncertainty of estimates and the effect of stochastic and uncertain time lags from data back to infections, is a major challenge. Our current strategy has been to report estimates of the reproduction numbers one week back in time as the most reliable estimates, and this needs to be studied further.
 As pointed out by one of the reviewers, it is quite easy to extend the modelling approach (and the algorithm) to include dummy variables describing interventions made by the government.
In this case, one would need to estimate a time delay between the time point in which the interventions are decided and when they effect viral transmission. This delay might change in time and be intervention specific. In our model, interventions appear in the data after a delay, and are then reflected in a change of the reproduction number. It is possible to interpret changes in the estimated reproduction numbers in the light of the interventions set in place.
Finally, it is interesting to compare the estimates of the instantaneous reproduction number produced by this SMC model, with the estimates obtained by models which keep the reproduction number constant over longer time intervals, for example over four weeks. The SMC-based \(R_t\)  is able to capture changes at shorter time scales significantly better, but possibly with larger uncertainty than estimates of reproduction numbers assumed constant over longer time periods, if the transmission has been stable during such periods. Comparing prediction power is a further aspects that can be compared.
Results from our SMC model are currently used in the weekly reports of the Norwegian Institute of Public Health, see www.fhi.no/en/publ/2020/weekly-reports-for-coronavirus-og-covid-19/.
